# Full text of "History Of The Theory Of Numbers - I"

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```284                   HISTORY OF THE THEORY OF NUMBERS.               [CHAP, x
where the exponents in the series are triangular numbers. Hence if we count the number of ways in which n can be formed as a sum of different terms from 1, 2, 3, ... together with different terms from 2, 4, 6, . . . , first taking an even number of the latter and second an odd number, the difference of the counts is 1 or 0 according as n is a triangular number or not. It is proved that
(10)
The fact that the second member must be an integer is generalized as follows: for n odd, a(n) is even or odd according as n is not or is a square; for n even, <r(n) is even if n is not a square or the double of a square, odd in the contrary case. Hence squares and their doubles are the only integers whose sums of divisors are odd.
V. Bouniakowsky20 proved that <r(AT)s2 (mod 4) only when N=kc2 or 2fcc2, where k is a prime 4Z+1 [corrected by Liouville30].
V. A. Lebesgue21 denoted by 1+A&+ A2x*+ . . . the expansion of the mth power of p(x), given by (1), and proved, by the method used by Euler for the case m = 1, that
This recursion formula gives
— 3)      A     — w(m— l)(m— 8) -t                            - 1